A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators

For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local best-approximation error of the finite element function by piecewise polynomial functions of the degree determining the expected approximation order, which need not coincide with the maximal polynomial degree of the element, for example if bubble functions are used. The error estimator is shown to be reliable and locally efficient up to this polynomial best-approximation error and oscillations of the right-hand side

Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D

We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}\Delta ^{2}u-\Delta u=f$ on the unit square $\Omega \subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial \Omega$, where $\varepsilon \in (0, 1)$ is a small parameter. The problem is solved numerically by means of a weak Galerkin(WG) finite element method, which is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on finite element partitions consisting of polygons of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with $N^2$ elements is constructed ,convergence of the method is proved in a discrete $H^2$ norm for the corresponding WG finite element solutions and numerical results are presented

Virtual element methods for fourth-order problems : implementation and analysis

In this thesis we aim to create a unified framework for the implementation and analysis of virtual element spaces. The approach we take for the virtual element discretisation allows us to easily construct vector field spaces as well as consider both variable coefficient and nonlinear problems. On top of this, the approach can be integrated more readily into existing finite element software packages. These are significant advantages of the method we present and something that has not been easy to achieve within the original virtual element setting. We exploit these key advantages in this thesis. In particular, we present a straightforward and generic way to define the projection operators, which form a crucial part of the virtual element discretisation, for a wide range of problems. We demonstrate how to build Hm-conforming for m = 1, 2 and nonconforming spaces as well as divergence and curl free spaces. All of which have been implemented in the open source Dune software framework as part of the Dune-Fem module. As a consequence of the projection approach taken in our framework, we are able to carry out a priori error analysis for higher order approximations of the following fourth-order problems: a general linear fourth-order PDE with non-constant coefficients; a singular perturbation problem; and the nonlinear time-dependent Cahn-Hilliard equation. Furthermore, we showcase the versatility of the projection approach with the introduction of a novel nonconforming scheme for the singular perturbation problem. The modified nonconforming method is uniformly convergent with respect to the perturbation parameter and unlike modifications in the literature, does not require an enlargement of the space. Numerical tests are carried out to verify the theoretical results